Differentiating Euler formula vs. multiplying by i

[ke7ijo]

TL;DR Summary

I ran into an apparent contradiction when working with Euler's formula and I can't find the mistake.

I differentiated both sides of Euler's formula with respect to x :
e^ix = sin x + i cos x => ie^ix = cos x - i sin x

Then for comparison I multiplied both sides of Euler's formula by i:
e^ix = sin x + i cos x => ie^ix = i sin x - cos x

Each of these two procedures seems to yield the additive inverse of the other, and I can't seem to figure out why even after a couple of hours of going back over it.

 

[Mark44]

I differentiated both sides of Euler's formula with respect to x :
e^ix = sin x + i cos x

The mistake is that you have misremembered Euler's formula. The correct version is $e^{ix} = \cos(x) + i\sin(x)$, which differs from what you wrote.

You can think of it this way. On the unit circle, with $x$ being the angle a ray makes with the horizontal axis, $e^{ix}$ represents the point on the unit circle. The coordinates of the point are $(\cos(x), \sin(x))$. As a complex number, this point is $\cos(x) + i\sin(x)$.

 

[ke7ijo]

Thank you! That's it.

 

[MidgetDwarf]

if you are mathematically curious, look at the different derivations of Euler's formula. My favorite is the one that uses Taylor Series.